HPC Sem 1 Study Guide

2C - Analyze graphs, know difference between functions and relations, find specific values from the graph of a function, state domain and range of function from a graph

-relations don't pass vertical line test -domain: range of x values -range: range of y values -f(2): plug in x for the x value and give the y value associated with it -f(x)=2: find the y value and give the x value associated with it

5B - Determine if a function is 1 to 1, find inverses, recognize relations between domain and range of a function and its inverse

1-1 if: passes vertical and horizontal line test. * Every x-value produces 1 unique y-value Typical examples: Usually work- linear, cubic Don't work- quadratic, absolute value, greatest integer Inverses To find: Switch x and y, then solve for y ** Only 1-1 functions can have inverses ***If you need to find the inverse of a non 1-1 function, find a point where you can turn it into an inequality Domain and Range of inverse functions are opposite each other. For example: f(x): domain is x>1 and y< -5 f inverse: domain is x< -5 and y>1

2E - Be able to sketch the following graphs; linear, quadratic, cubic, square root, cube root, absolute value, reciprocal, greatest integer, least integer, and be able to graph piecewise functions

1. Linear: straight line in any direction 2. Quadratic: parabola, up or down 3. Cubic: line w/ one squiggle, up or down 4. Absolute value: V graph 5. Square root: half parabola, side to side 6. Cubic: line w/ one squiggle, side to side 7. Reciprocal: two (or more) curved lines 8. Greatest Integer: steps Translation: x-- f(x-h) : horizontal h units y-- f(x) + k : vertical k units Dilation: x-- f(h*x) : horizontal by factor of 1/h y-- k*f(x) : vertical by factor of k Absolute Value: x -- f(IxI) : reflection across the y axis y -- If(x)I : all negative y-values become positive *If struggling: just plug in some values For piecewise functions: Graph as usual BUT **remember the vertical line test!!

12A - Find the first n terms of a sequence, evaluate recursive sequences, evaluate expressions that contain a sigma notation

A sequence is a function whose domain is the natural numbers. Defining recursively: the sequential terms can be defined by previous terms. For arithmetic sequences: An = A1 + d(n-1) where d is common difference For geometric sequences: An = A1 + r^n-1 summation notation: -on bottom k = # is the starting point -on top n is the ending point -to the right is the sequence, replace k for the term you are looking for

12E - Analyze various types of series, inluding A/G and telescoping series

A/G series: arithmetic/geometric 1. Multiply by the common ratio of the denominator 2. List out first few terms and add the initial sum and the sum multiplied by a constant 3. Solve as a system of equations 4. Note and solve for the new sum which is a geometric sequence **1st term of sum multiplied by a constant may not fit the rest Telescoping Series -Series that include quadratics (or polynomials) 1. Separate sequence into a partial fraction decomposition (PFD) 2. List out the sum of the first however many terms until fractions start to cancel out 3. Add the remaining fractions to get the sum. **For cubic telescoping series, when canceling out terms, you may add two terms that cancel out one other one, ie. -1/3 -1/3 +2/3, you can cancel them all out

5E - Solve exponential and logarithmic equations

Choose method of best fit: - logb(a)= x >> b^x = a - change of base, logb(a) = logc(a)/logc(b) - clogb(a) = logb(a)^x - a^(loga(m) = m - log(m) -log(n) = log (m/n) - log(m) + log(n) = log(m*n) Exponential: - take log of both sides - separate exponents, break it down

12B - Analyze arithmetic sequences

Common difference is difference between terms. to find the value for a certain term or the number of a given term: An = A1 + d(n-1) sum of an arithmetic sequence: S = (A1 +An)/2 * n **You may have to set up a system of equations if you are given two sets of unknowns ***Usually best to solve for common differences, for example: instead of starting: A1 + A23 = 100 do: A1 + 22d =100, then you have two common variables

5F - Simply expressions involving logarithms and exponents

Use: - log(a) + log(b) = log(ab) to solve going backwards, or to expand - log(a) - log(b) = log(a/b) to solve going backwards, or to expand

12C - Analyze geometric sequences, find sum of both finite and infinite series

Common ratio is the ratio between terms. aka b/a = c/b to find the value for a certain term or the number of a given term: An = A1 * r(n-1) Sum of a finite geometric series: S = A1 * (1 - r^n)/(1-r) Sum of infinite geometric series (if IrI < 1) S = A1/1-r **You may have to set up a system of equations if you are given two sets of unknowns ***Usually best to solve for common ratios

5A - Find composite of 2 functions, including stating domain

Composition of Functions: An operation on 2 functions (f of g)(x) is like f(g(x)) -- go right to left 1. do g of x 2. plug in g(x) to all of the x values in f(x) 3. find the domain by union of domain of g(x) and domain of f(g(x)) When the function is a set of points: if it's f(g(x): 1. Make a 3 column chart 2. Using two columns list the points in g(x), both the x and y- values 3. Use the g(x) values (y- values) and plug them into the domain of f(x) , find the output for each value 4. Your new function is the input of g(x) and the corresponding output of f(x)

3A - Analyze quadratic functions and relations, finding vertex, intervals of increasing/decreasing, x + y intercepts, and sketch the graphs, solve equations involving quadratic inequalities

Definition: A polynomial function of degree 2 Forms: - Standard: f(x) = Ax^2 + Bx + C , A can't = 0 - Vertex: f(x) = A(x-h) + k >> vertex is (h,k) - Factored: f(x) = A(x-r1)(x-r2) >> r1 and r2 are x intercepts To find: X- intercepts: factor or quadratic formula Y- intercepts: f(0) or C in standard form Vertex: complete the square To find # of x-intercepts: Discriminant: B^2 - 4AC For inequalities: Always factor! Never divide! Then graph the roots, and find the direction by its sign

11B - Solve systems of equations/matrices using Creamer's Rule and determinants

Determinants: -ad - bc for 2 by 2 -multiply diagonals forward, add the three going right, multiply diagonals backward, add the three going left, subtract, for 3 by 3 -Cramer's Rule: value of x equals Dx/D -To find Dx, replace x column with solution values Row operations effect on determinants - Switching rows: Opposite determinant - Multiply row by constant: Determinant is multiplied by the constant - Add to a multiple of a row: Determinant is unchanged

4B - Analyze rational functions, including stating domain, identifying vertical, horizontal, and oblique asymptotes, finding x and y intercepts, holes, sketch graph

Domain: look at denominator for values that make it zero, or a hole is made when there is a factor that is the same in the numerator and the denominator Vertical Asymptote: at x=r if (x-r) is factor of the DENOMINATOR Horizontal Asymptote: -degree of numerator < deg of denom: y=0 -deg of num = deg of den: y= coefficient of num/coefficient of den Oblique Asymptote: -deg of num > deg of den: quotient using synthetic division (ignore remainder) Finding Interceps X-intercept: the value that makes the denominator = 0 Y-intercept: when x = 0 NOTE THIS ** Functions can cross horizontal asymptotes, HAs only matter at far left and right -- make sure to test HAs to see if they cross by setting y = # equal to function ** When a vertical asymptote has an odd multiplicity, the function starts up again on opposite end (y is very pos or very neg) ** When a vertical asymptote has an even multiplicity, the function starts up again on the same end (y is very pos and y is very pos, or reverse)

2D - Determine if function is even or odd or both, determine if a function is increasing, decreasing or constant, use calculator to find local and absolute maximums or minimums

Even function: f(x) = f(-x) , symmetrical across the y-axis, if (x,y) then (-x,y) Odd function: f(x) = -f(x), rotate 180, symmetry about origin, if (x,y) then (-x,-y) **To test select a value and plug them in for f(x), f(-x) and -f(x) to see if they are the same To locate mins and max on Calculator: 1. Graph the function 2. Make sure window is large enough 3. 2nd trace - max.min 4. place left bound and right bound and guess

10A - Analyze parabolas

General Form - opening left or right: (y-k)^2=4a(x-h) - opening up or down: (x-h)^2=4a(y-k) **Where (h,k) is the vertex focus: a fixed point directrix: a fixed line **Parabola is a collection of all points that are equdistant from the focus and the directrix latus rectum = the line segment that passes through the focus and is parallel to the directrix 4a = length of the latus rectum a = distance from vertex to focus or directrix 2a = distance from focus to endpoints of the latus rectum

10D - Analyze the 4 conic sections

General Formula Ax^2 + Bxy + Cy^2 + Dx + Ey + F= 0 Parabola: A or C = 0 Ellipse: A and C are the same sign Hyperbola: A and C are opposite signs Eccentricity: e= c/a For 0<e<1 : ellipse For e = 1: parabola FOr e> 1: hyperbola The ratio between the distance to a fixed point and a fixed line is e **if a fraction, numerator is distance to point, denominator is distance to line

10C - Analyze hyperbolas

General Formulas -go up and down: (x-h)^2 (y-k)^2 _______ - _______ = 1 a^2 b^2 -go side to side: (y-k)^2 (x-h)^2 _______ - _______ = 1 a^2 b^2 c^2= a^2 + b^2 a is the distance from the vertices to the center b is the distance from the center to the ends of the box c is the distance from the center to the foci (on the transverse axis) -transverse axis passes through points (and center) -conjugate axis doesn't pass through points, but through center

10B - Analyze ellipses and circles

General Formulas -larger horizontally: (x-h)^2 (y-k)^2 _______ + _______ = 1 a^2 b^2 -larger vertically: (x-h)^2 (y-k)^2 _______ + _______ = 1 b^2 a^2 b^2=a^2-c^2 a and b are distances to vertices from the center c is the distance to the foci (major axis) from the center

11A - Solve matrices by using row operations

IMPORTANT - Get a 1 in the top left corner first - Then get 0s left side of the other rows - Get a 1 in the second column - Get 0s in the rest of the rows of the second column - Repeat for how many columns there are 3 Row Operations - Switch two rows - Add one row to the multiple of another - Multiply one row by a constant Consistent: Has solutions #Independent has 1 solution #Dependent has infinite solutions Inconsistent: No Solutions CALC TUTORIAL (solving matrices): 1. 2nd. Matrix - Edit 2. Enter data 3. 2nd quit 4. 2nd Matrix- Math- Rref 5. 2nd Matrix [A]- Enter

4C - Solve equations involving polynomial and rational inequalities

Make a line graph and plot all x-intercepts, and horizontal asymptotes and holes. Take note of end behavior and sketch it on your line graph. Use the multiplicity of each x-intercept and determine the direction it continues in. Use equation to solve for the inequality, making sure to account for openings from vertical asymptotes or holes

3B - Create quadratic functions based on mathematical models, calculate equation of best fit using a calculator of linear and quadratic data

Mathematical Models 1. Find the purpose and place of each number given to you in the word problem 2. Find the purpose and value that you need to find 3. Substitute and solve Line of best fit 1. Stat - edit - enter data 2. go to the y= menu and select plot 1 3. graph 4. Stat - calc - quadreg

11C - Add, subtract and multiply matrices as well as find inverses

Matrix contains m rows, and n columns, dimensions m x n -ADDITION (or subtraction): Add individual values in corresponding locations -SCALAR MULTIPLICATION (multiplying by a constant): Multiply each individual value in ever location -MATRIX MULTIPLICATION (only works with dimensions of m x n and n x r): Each spot filled by multiplying corresponding row of first matrix with corresponding column of second matrix and adding each product -DIVISION: Multiplication of inverses. To divide A/B is the same as multiplying A by B inverse. TO find inverse: use row operations to transform the matrix [A,I] to [I,A inverse]. **Note: only square matrices can have multiplicative inverses. Doesn't mean all square matrices do. Nonsingular matrices have inverses, singular matrices do not. IF MATRIX HAS DETERMINANT OF 0, IT HAS NO INVERSE Matrix of [I] is just 1s in diagonal going from top left corner to bottom left corner. 0s in all other locations. - Dependent vs. Inconsistent difference: if Dx is equal to 0, it is dependent if Dx is not 0, it is inconsistent

5D - Graph logarithmic functions without a calculator

Properties - A is the base, a is greater than 0, but not 1 - x is the argument, has to be > 0 - Vertical, crosses horizontal axis - D: x > 0 - R: y: all reals - X - intercept: (1,0) - No y intercepts (unless (x-h) in argument) - **Critical point >> when argument = 1 To find x-intercepts: set y = 0 y-intercepts (when function is shifted left): set x = 0 critical point: set argument = 0

5C - Graph exponential functions without a calculator

Properties: - f(x) = a^x - a is base; a is greater than 0 but not 1 - D: all reals - R: y> 0 - Horizontal; crosses y-axis - No x-intercepts - Y-intercept is 1 - Horizontal asymptote is y = 0 To find X intecerpts (must have a negative k value): - set y equal to O, usually gonna be a log Y intercepts: - Set x equal to zero

2A - Analyzing functions, difference between functions and relations, find specific values of functions, finding domain and range of functions

Relation: Correspondence between elements of two sets Function: Relation where an element of one set (domain) corresponds with at most one element of the other set (range) Finding specific values: plug in the number to the equation for x Finding domain and range: make sure denominator can't equal zero

4D - Find complex zeros of a polynomial equation by; potential rational roots, remainder theorem, synthetic division, and conjugate pairs

Remainder Theorem: If f(x) is divided by (x-c), then the remainder is f(c) *Basically plug c into to equation Factor Theorem: (x-c) is a factor of f(x) IF f(c)=0 Descartes' Rule of Signs: -Positive Real Zeros: Number of sign changes of f(x) or else less an even integer -Negative Real Zeros: Number of sign changes of f(-x) or else less an even integer Potential Rational Roots Theorem: F has potential roots at p/q, where p = all integral factors of the constant term and q = all integral factors of the leading coefficient (last term int/first term int) ** solve by synthetic division or quadratic formula for complex solutions **Pay attention to upper and lower bounds Upperbound: when the coefficients of the solution continue increasing Lowerbound: when the coefficients of the solution alternate positive and negative

4A - Analyze polynomials, sketching graphs by hand, finding real zeros, creating equations based on graphs

STEPS 1. Determine end behavior by finding degree (odd-split, even- same direction) and sign (+ ends up, - ends down) 2. Find x and y intercepts by plugging in 0 to the equation 3. Find the zeros and their multiplicites. (if odd, goes through, if even, just bounces) 4. Approximate turning points (a polynomial has n-1 turning pts, or and even # less) 5. Use information from steps 1-4 to graph by hand 6. Find the domain and range of function 7. Figure out where function is increasing and decreasing

11D - Decompose rational expressions into partial fractions

Steps 1. Factor the denominator 2. Separate into it's factors - If it has degree of 1 the numerator is A - If it has an irreducible quadratic factor, numerator is Ax +B 3. Multiply out. A by other denominators, etc. - Should get something like: Ax^2 + 2Ax +Bx^2 + 3B / CD 4. Set up a system of equations - for x^2: A + B = (# of square values in original numerator) - for x: 2A = (# of x values in original numerator) - for #: 3B = (# of constant in original numerator) 5. Plug back into equation for the PFD

12D - Use induction to prove algebraic statements

Steps 1. Verify S1 >> (substitue 1 for n) 2. Assume Sk >> (substitute k for n) 3. Verify Sk+1 >> (subsititue k +1 for n) then substitute first equation into first part of second equation. Example: 1+5+9...(4n-3) = n(2n -1) 1. 1(2-1) = 1 2. Assume 1+5+9+....+(4k-3) = k(2k-1) 3. Verify: 1+5+9+....+(4k-3) + (4(k+1) -3) = (k+1)(2(k+1) -1) k(2k-1) + 4k +1 = (k+1)(2(k+1) -1)

2B - Find sum, difference, product and quotient of two functions, find the difference quotient of a function, calculate average rate of change of a function

Sum: (f+g)(x) = f(x) + g(x) -- just add them together Subtraction: (f-g)(x) = f(x) - g(x) -- just subtract Multiplication: (f*g)(x) = f(x)g(x) -- just multiply Division: (f/g)(x) - f(x)/g(x) -- just divide Difference Quotient: f(x+h)-f(x)/h * plug in (x+h) for all x's for first part, then subtract original function then divide by h Average rate of change: slope, rise over run, between 2 values is the slope between two points of a function (slop of a secant line). * plug in the values given into the equation for the numerator, and the given values in the denominator, change in y over change in x **Note: If you perform an operation on 2 functions, the domain is the intersection of the domains of the components, minus any other values, due to the result.

5G - Solve exponential growth and decay problems, calculate exponential, logarithmic and logistic curve of best fit using a caluclator

Uninhibited Growth and Decay Formulas: Growth/Decay Model: A(t) = A(base 0)(times e to the k times t) -where if k<0 A is decreasing and if k> o A is increasing Logistic Model (acknowledges there is a limit): P(t) = c/ 1 + ae^-bt ON CALC: LOGISTIC Logarithmic Model (steep then flat) y = a + b(ln)x ON CALC: LNREG Exponential Model (flat then steep) y = a * b^x ON CALC: EXPREG **You can convert this to growth formula by finding the natural log of b : e^k= b, so k= #, then you have e^kt